Generalizations of Left $\varphi$-Biflatness and Existence of Module Means on ‎Banach Algebras‎‏

Abstract

Motivated by recent developments in the homological theory of Banach modules, we introduce and study the notion of module left $(\phi,\varphi)$-biflatness in the setting of Banach $\mathfrak{A}$-bimodules. Alongside this concept, we define suitable module $(\phi,\varphi)$-means on associated dual and bidual spaces, which provide a natural tool for analyzing this biflatness property. We show that, under mild assumptions on quotient modules, module left $(\phi,\varphi)$-biflatness can be characterized in terms of the existence of module $(\phi,\varphi)$-means. Furthermore, we establish that this biflatness property implies a corresponding module amenability-type condition on quotient algebras. Finally, we prove that module left $(\phi,\varphi)$-biflatness is stable under split extensions of Banach $\mathfrak{A}$-bimodules, thereby providing a method for constructing new examples and extending known results.